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The tri-hexaflexagon

Watch to learn how to fold a basic paper flexagon
So, we’re going to make our first flexagon, the tri-hexa-flexagon. It’s made out of a straight strip of triangles. You can download the template below, and we’ll make the flexagon together. Print out and glue together the back and front, so that the paper is printed on both sides. Make sure you do this the right way, so that one side looks like this… and when you turn it over from top to bottom … the other side looks like this. The first thing we want to do is to fold the paper along the diagonal lines, back and forth, so that the paper is flexible and easy to work.
So we’re creasing back and forth, like this, just so that later along it’s easier to fold and flex. Now, we have the paper strip ready. The tri-hexa-flexagon is shaped as a hexagon and has three faces. One of the key secrets to folding flexagons, is that you always fold together triangles with the same colour, face to face. So I’m going to fold, yellow to yellow, and we get a hexagon with a flap, rather like an envelope. The flap is made out of these two white triangles, which you simply tape or glue together. And that’s it! There’s your tri-hexa-flexagon. At the moment you can see two faces, the red face and the blue face.
Now, let’s learn how to flex the flexagon, using what’s known as the most basic manoeuvre, the pinch flex, to reveal the third yellow face. The way to flex, is to fold the flexagon like this, making the shape of a three-pointed propeller. Now, you put your finger at the point in the middle, and you should find that it opens up to reveal the third face, like this. However, about half of the time, you’ll find that there is no opening at this point. This is because, if you look carefully at the spread out flexagon, there are only 3 open pockets.
All you need to do to correct this, is to change all the mountain folds to valley folds, and all the valley folds to mountains, like this. Now the flexagon should open up at the top. The tri-hexa-flexagon is cyclic. You can go on and on, flexing it forever, revealing one after the other the three faces, red, yellow and blue.

My first flexagon

This is your first experience with folding a flexagon. The way we’re going to work in this course is as follows:

  • Download the file below which has the templates for the back and front of the paper strip that will be folded into a tri-hexa-flexagon.
  • Print the two templates using a colour printer.
  • Glue together the back and the front, so that the paper strip is printed on both sides. Make sure you do this the right way, and that your back and front are correctly positioned. The order of the coloured triangles on one side from left to right should be red, blue, blue, yellow, yellow, red, red, blue, blue, yellow; On the other side, when you turn over the strip from top to bottom you should have, from left to right: white, yellow, red, red, blue, blue, yellow, yellow, red, white.
  • Crease the paper along the diagonal lines, back and forth, so that the paper is flexible and easy to work.
  • Fold the paper strip into a hexagon with the two white triangles creating a ‘flap’. Make sure you fold same-coloured triangles together. Glue or tape the flaps to each other, and that’s it!

You’ll be able to see two faces or sides of the flexagon, once it’s assembled. The blue and the red sides. To expose the third face, you’ll have to do the pinch flex, as shown in the video. This might be a bit tricky at first, but please don’t give up! You’ll get it in the end.

Pats and Leaves

One common mistake happens when the three-pointed propellor shape doesn’t open from the center. To remedy this all you have to do is switch the directions of all the folds between the triangles. Mountain folds should be changed to valley folds and vice-versa. This happens because of the basic structure of the flexagon. Rotate the flexagon in your hand moving from triangle to triangle. You’ll find that the thickness of the flexagon alternates between stacks of either 1 or 2 ‘leaves’. These stacks of leaves are called ‘pats’. So, this flexagon has 6 pats, alternating between 1 or 2 leaves thick as we rotate the flexagon in our hand. The number of the leaves (the visible side of them) on a face is equal to the number of pats.

Fun with the tri-hexa-flexagon

After a bit of practice you’ll be fluent and will be able to cycle through the 3 faces of the flexagon over and over again. You might want to start exploring your flexagon right away. Try attaching a paper clip to one of the triangles and see how it circles around the flexagon in different positions. If you draw some lines or dots on all of the triangles on all of the faces, you’ll be able to see some pretty patterns emerging, once you flex. Draw a straight line from the centre of the flexagon to the edge of each triangle so that it bisects the angle. Flex the flexagon! What happens to the line?


Did you enjoy folding the flexagon? Did you have difficulties? Did you discover tricks that you would like to share with the other learners?

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Flexagons and the Math Behind Twisted Paper

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